Calculus I (MAT 145) Dr. Day Monday Oct 23, 2017 Derivative Shortcuts (Chapter 3) Related Rates (3.9) Return Quiz #9 Monday, October 23, 2017 MAT 145

Position, velocity, acceleration Velocity: rate of change of position Acceleration: rate of change of velocity Velocity and acceleration are signed numbers. Sign of velocity (pos./neg.) indicates direction of motion (right/left or up/down) When velocity and acceleration have the same sign (both pos. or both neg.), then object is speeding up. This is because object is accelerating the same direction that the object is moving. When velocity and acceleration have opposite sign (one positive and one negative), then object is slowing down. This is because object is accelerating the opposite direction that the object is moving. Monday, October 23, 2017 MAT 145

Implicit differentiation with respect to another variable Suppose x, y, and z are RELATED in the following way: And, furthermore, suppose that EACH of x, y, and z are RELATED, in some unknown way, to another VARIABLE, t: How are each of x, y, and z changing, WITH RESPECT TO t? Monday, October 23, 2017 MAT 145

Related Rates Suppose a right triangle is growing over time. We can implicitly differentiate an equation that relates the measures of the sides to find a relationship between the rates at which the sides are changing. dx/dt represents the rate at which the side labeled x is changing with respect to time. dy/dt represents the rate at which the side labeled y is changing with respect to time. dz/dt represents the rate at which the side labeled z is changing with respect to time. Monday, October 23, 2017 MAT 145

Enlarging Circle A stone is dropped in a pond and a circle ripples out from that point. The first circle’s radius increases at a rate of 2 cm/s. How fast is the area of the circle increasing when the radius is 30 cm? Vars: r, A; Given: dr/dt = 2 cm/s; Requested: dA/dt, when r=30 cm Relationship between variables Implicit differentiation with respect to time t leads to… Relationship between rates: Related Rates! Monday, October 23, 2017 MAT 145 Ripples on the Pond

Enlarging Circle A stone is dropped in a pond and a circle ripples out from that point. The first circle’s radius increases at a rate of 2 cm/s. How fast is the area of the circle increasing when the radius is 30 cm? Interpret: The area of the first circle is increasing at the rate of 120π cm2/sec, when the radius of the first circle is 30 cm. Related-rates equation Substitute known values for rates and variables Solve for desired rate Monday, October 23, 2017 MAT 145

Growing balloon Air is being pumped into a spherical balloon so that its volume increases at a rate of 100 cm3/s. How fast is the radius of the balloon increasing when the diameter is 50 cm? Vars: r, V; Given: dV/dt = 100 cm3/s; Requested: dV/dt, when d = 50 cm (or r=25 cm) Relationship between variables Implicit differentiation with respect to time leads to… Relationship between rates Monday, October 23, 2017 MAT 145

Growing balloon Air is being pumped into a spherical balloon so that its volume increases at a rate of 100 cm3/s. How fast is the radius of the balloon increasing when the diameter is 50 cm? Interpret: The radius of the balloon is increasing at 1/1ooπ cm/s, when the diameter of the balloon is 50 cm. Related rates equation Substitute known values for rates and variables (Note: d=2r) Solve for desired rate Monday, October 23, 2017 MAT 145

Ladder problem A ladder 10 ft long rests against a vertical wall. If the bottom of the ladder slides away from the wall at a rate of 1 ft/s, how fast is the top of the ladder sliding down the wall when the bottom of the ladder is 6 ft from the wall? Monday, October 23, 2017 MAT 145

Conical water tank A water tank has the shape of an inverted cone with a base radius 2 m and height 4 m. If water is being pumped into the tank at 2 m3/min, find the rate at which the water level is rising when the water is 3 m deep. Monday, October 23, 2017 MAT 145

Two cars heading toward intersection Car A is traveling west at 50 mi/hr and car B is traveling north at 60 mi/hr. Both are headed for the intersection of the two roads. At what rate are the cars heading for each other when car A is 0.3 mi and car B is 0.4 mi from the intersection? Note: Read “rate cars heading for each other” as “rate at which direct line distance between them is decreasing.” Monday, October 23, 2017 MAT 145

Related-Rates Strategy Identify variables, given values and rates, and requested rate at a specific moment in time (perhaps indicated by specific value of variable). Relate given variables. Implicitly differentiate to find relationship among the rates. If necessary, solve associated problem to find needed auxiliary relationships. Substitute and solve for requested rate at specific time. Report and interpret requested rate as increasing/decreasing and provide units! Monday, October 23, 2017 MAT 145

Related Rates: Assessing Your Work Define all variables, by explaining what each variable you are using represents. Include a sketch, labeled with variables or algebraic expressions that include variables. Identify all rate(s) included in this problem, and, if known, provide their values, and their units. Show full evidence of your calculus-based reasoning and steps to solve the problem, including calculations or calculator input and output. Write a sentence to interpret the results of your calculations, including units.   __ 1 pt: related sketch, drawing, graph; __ 1 pt: related-variables equation; __ 1 pt: related-rates equation; __ 1 pt: use known info; __ 3 pts: correct & correctly labeled result, with units, written as a sentence written as a sentence labeled result, with units, ____ 3 pts: correct and correctly ____ 1 pt: use known info; ____ 1 pt: related-rates equation; ____ 1 pt: related-variables equation; ____ 1 pt: related sketch, drawing, graph;   /7 TOTAL

Related Rates (3.9) Illustrations and Examples Temple University RR site (discussion and animation) Kelly’s RR notes (solutions and animations) Melting Snowball discussion and animation) More Temple University animations Paul’s Online Math Notes: RR (worked examples) More Relatives RR (worked examples) Monday, October 23, 2017 MAT 145